Let us comment on the role of the three quantities \(\mathrm {vol}(Y), \mathrm {inj}(Y)\)  and \(\lambda _1^*(Y)\)  that appear in the statement of the theorem. As mentioned above, there are only finitely many hyperbolic three-manifolds with \(\mathrm {vol}(Y)\le V\)  and \(\mathrm {inj}\ge \varepsilon \) . Of course, this finiteness properties implies that there is a lower bound on \(\lambda _1^*(Y)\)  in such set of generators in terms of \(V,\varepsilon \) . The problem is that there is no known explicit bound in terms of those quantities. The bounds provided in [1]} take as input significantly more information, namely the volume and a sufficiently large portion of the complex length spectrum of \(Y\) ; furthermore, they involve an optimization procedure that makes them implicit in nature.
